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Two-dimensional conformal field theory plays a fundamental role in the theory of two-dimensional
critical systems of classical statistical mechanics, in quasi one-dimensional condensed matter
physics and in string theory. The study of defects in systems of condensed matter physics, of
percolation probabilities and of (open) string perturbation theory in the background of certain
solitonic solutions of string theory, the so-called D-branes, forces one to analyze conformal field
theories on surfaces that may have boundaries and/or can be non-orientable. This study has
recently led to much new insight in the mathematical structure of conformal field theory.
Many mathematical disciplines have contributed to a better understand of conformal field
theory and received stimulating input from questions arising in conformal field theories.

There
are two major approaches to chiral conformal field theory: one that is based on operator
algebras and one based on vertex algebras. Both structures lead to representation categories
that are tensor categories and, in the case of rational chiral conformal field theories, more
specifically modular tensor categories. They also encode the monodromy representations of
the vector bundles of conformal blocks for rational vertex algebras, objects that are of interest
for algebraic geometry. Moreover, modular tensor categories are a crucial ingredient in the
construction of three-dimensional topological field theories.

While chiral conformal field theories have certain physical applications in the description of
quantum Hall systems, full local conformal field theories are relevant for the physical applications
referred to in the first paragraph. Recently, it has been understood that the construction
of a full local conformal field theory is bestdescribed using the structure of a module category
over the tensor category that describes the chiral data.

In this Arbeitsgemeinschaft, we will explain this mathematical notion, related concepts and
some applications. The category theoretic framework allows to emphasize those aspects that
are common to all approaches to chiral conformal field theory.

Posted at January 29, 2007 6:16 PM UTC

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1137

Read the post Oberwolfach CFT, Tuesday MorningWeblog: The n-Category CaféExcerpt: On Q-systems, on the Drinfeld Double and its modular tensor representation category, and on John Roberts ideas on nonabelian cohomology and QFT.Tracked: April 3, 2007 2:06 PM